Digital communications - Codes and Turbo Codes - page 37

chosen for the GSM (Groupe Spécial Mobile and later Global System for Mobile
communications ) system. We note that there is no simple expression of the
power spectral density of a GMSK signal. For values of the normalized passband
of 0.3 or of 0.2, the power spectral density of the GMSK signal does not show
sidelobes and its decrease as a function of frequency is very rapid. Thus at -10
dB the band occupied by the GMSK signal is approximately 200 kHz, and at
-40 dB 400 kHz for a rate D = 271 kbit/s.

2.2 Structure and performance of the optimal re-

ceiver on a Gaussian channel

The object of this chapter is to determine the structure and the performance
of the optimal receiver for memory and memoryless modulations on an additive
white Gaussian noise (AWGN) channel. The type of receiver considered is the
coherent receiver where the frequency and the phase of the signals transmitted
by the modulator are assumed to be known by the receiver. Indeed, a coherent
receiver is capable of locally generating signals having the same frequency and
the same phase as those provided by the modulator, unlike the so-called non-
coherent or differential receiver .
Generally, the receiver is made up of first a demodulator, the aim of which is
to convert the modulated signal into a baseband signal, and second, a decision
circuit in charge of estimating the blocks of data transmitted. The receiver
is optimal in the sense that it guarantees a minimal error probability on the
estimated blocks of data.

2.2.1 Structure of the coherent receiver

Let s j ( t ) , j =1 , 2 , ,M be the signals transmitted on the transmission chan-

nel perturbed by a AWGN b ( t ) , with zero mean and of power spectral density
equal to N 0 / 2 . On the time interval [0 ,T [ , the signal received by the receiver is
equal to:

···

r ( t )= s j ( t )+ b ( t )
The Ms j ( t ) signals define a space of dimension N

M , and can be represented
in the form of a series of normed and orthogonal weighted functions ν p ( t ) .

≤

N

s j ( t )= s jp ν p ( t )

p =1

where s jp is a scalar equal to the projection of the signal s j ( t ) on the function
ν p ( t ) .

T

s jp = s j ( t ) ν p ( t ) dt

0